how to use a driver to drive object on object's local axis
AFAIK, you have two options:


Assign a Transform constraint to the object. This allows you to map a transform range of the source object to a transform range of the target object, in either their World or Local spaces.
More conventionally, parent the driven object to another one (often an Empty). Then the driven object's transform will be evaluated in its parent's space, in a driver expression. If the parent is aligned to the child when the relation is formed, that will be equivalent to the child's Local Space